Problem: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}10x+11y-12z&=-18 \\6x-8y-2z&=10 \\4x-6y-8z&=12\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Solution: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}10x+11y-12z&=-18 \\6x-8y-2z&=10 \\4x-6y-8z&=12\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{10}x+{11}y+({-12})z&=-18 \\{6}x+({-8})y+({-2})z&=10 \\{4}x+({-6})y+({-8})z&=12\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {10} & {11} & {-12} \\ {6} & {-8} & {-2} \\ {4} & {-6} & {-8} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {10} & {11} & {-12} \\ {6} & {-8} & {-2} \\ {4} & {-6} & {-8} \end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} -18 \\ 10 \\ 12 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 10 & 11 & -12 \\ 6 & -8 & -2 \\ 4 & -6 & -8 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} -18 \\ 10 \\ 12 \end{array} \right]$